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Description:
Space efficient data structures for partial ordered sets or posets are well-researched field. It is known that a poset with n elements can be represented in n²/4 + o(n²) bits [Munro and Nicholson, 2016] and can also be represented in (1 + ε)n log n + 2nk + o(nk) bits [Farzan and Fischer, 2011] where k is width of the poset. In this paper, we make the latter data structure occupy 2n(k-1) + o(nk) bits by considering topological labeling on the elements of posets. Also considering the topological labeling, we propose a new data structure that calculates queries on transitive reduction graphs of posets faster though queries on transitive closure graphs are computed slower. Moreover, we propose an alternative data structure for topological labeled posets that calculates both of the queries faster though it uses 3nk - 2n + o(nk) bits of space. Additionally, we discuss the advantage of these data structures from the perspective of an application for BlockDAG, which is a more scalable version of Blockchain.